#### Volume 11, issue 3 (2011)

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Polynomial $6j$–symbols and states sums

### Nathan Geer and Bertrand Patureau-Mirand

Algebraic & Geometric Topology 11 (2011) 1821–1860
##### Abstract

For a given $2r$–th root of unity $\xi$, we give explicit formulas of a family of $3$–variable Laurent polynomials ${J}_{i,j,k}$ with coefficients in $ℤ\left[\xi \right]$ that encode the $6j$–symbols associated with nilpotent representations of ${U}_{\xi }\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$. For a given abelian group $G$, we use them to produce a state sum invariant ${\tau }^{r}\left(M,L,{h}_{1},{h}_{2}\right)$ of a quadruplet (compact $3$–manifold $M$, link $L$ inside $M$, homology class ${h}_{1}\in {H}_{1}\left(M,ℤ\right)$, homology class ${h}_{2}\in {H}_{2}\left(M,G\right)$) with values in a ring $R$ related to $G$. The formulas are established by a “skein” calculus as an application of the theory of modified dimensions introduced by the authors and Turaev in [Compos. Math. 145 (2009) 196–212]. For an oriented $3$–manifold $M$, the invariants are related to $\tau \left(M,L,\varphi \in {H}^{1}\left(M,{ℂ}^{\ast }\right)\right)$ defined by the authors and Turaev in [arXiv:0910.1624] from the category of nilpotent representations of ${U}_{\xi }\left(\mathfrak{s}\mathfrak{l}\left(2\right)\right)$. They refine them as $\tau \left(M,L,\varphi \right)={\sum }_{{h}_{1}}{\tau }^{r}\left(M,L,{h}_{1},\stackrel{̃}{\varphi }\right)$ where $\stackrel{̃}{\varphi }$ correspond to $\varphi$ with the isomorphism ${H}_{2}\left(M,{ℂ}^{\ast }\right)\simeq {H}^{1}\left(M,{ℂ}^{\ast }\right)$.

##### Keywords
$6j$–symbols, state sum, skein calculus, quantum groups, $3$–manifolds
##### Mathematical Subject Classification 2000
Primary: 57M27, 81Q99